This question tests 4th grade ability to decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation (CCSS.4.NF.3.b). Decomposing a fraction means breaking it apart into a sum of smaller fractions—like taking 8/12 and writing it as 5/12 + 3/12, or 2/12 + 2/12 + 4/12, or 4/12 + 4/12. The key requirement: all fractions in the decomposition must have the SAME denominator (all twelfths, etc.), and the numerators must add up to the original numerator. There are many different ways to decompose the same fraction, as long as these rules are followed. To decompose 8/12, students must write it as a sum of fractions with denominator 12, where the numerators add to 8. Different decompositions show different ways to partition the 8 parts: 8 twelfths can be grouped as 5+3, or 2+2+4, or 4+4, etc., all representing the same total. Choice A is correct because both decompositions use denominator 12 throughout, the numerators in first decomposition add to 8: 5 + 3 = 8, the numerators in second decomposition add to 8: 2 + 2 + 4 = 8, and the two decompositions are different from each other (different number of addends). This demonstrates understanding that fractions can be decomposed multiple ways while maintaining same denominator and total value. Choice B represents using different denominators, which happens when students don't maintain same denominator requirement or use equivalent fractions incorrectly. To help students: Use visual models (area models, fraction bars) to show different ways to group the same total—8/12 shaded in a rectangle can be grouped as (5 red + 3 blue)/12 or (2 red + 2 green + 4 blue)/12, same total but different groupings. Emphasize SAME DENOMINATOR always—when decomposing 8/12, every fraction must have denominator 12 (all twelfths); check that numerators sum correctly: 5 + 3 = 8 ✓, 2 + 2 + 4 = 8 ✓; connect to addition fact families: if you know 5 + 3 = 8 and 2 + 2 + 4 = 8, you can decompose 8/12 those ways; show that order doesn't matter (5/12 + 3/12 = 3/12 + 5/12), so don't count these as two different ways; watch for: changing denominators, numerators that don't sum correctly, repeating same decomposition, and counting commutative versions as different.

This question tests 4th grade ability to decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation (CCSS.4.NF.3.b). Decomposing a fraction means breaking it apart into a sum of smaller fractions—like taking 3/4 and writing it as 1/4 + 2/4, or 1/4 + 1/4 + 1/4. The key requirement: all fractions in the decomposition must have the SAME denominator (all fourths), and the numerators must add up to the original numerator. To decompose 3/4, students must write it as a sum of fractions with denominator 4, where the numerators add to 3. Different decompositions show different ways to partition the 3 parts: 3 fourths can be grouped as 1+2, or 1+1+1, etc., all representing the same total. Choice A is correct because both decompositions use denominator 4 throughout, the numerators in first decomposition add to 3: 1+2=3 ✓, the numerators in second decomposition add to 3: 1+1+1=3 ✓, and the two decompositions are different from each other (one uses two fractions, the other uses three). Choice B represents using different denominators (1/2 has denominator 2, not 4), which happens when students don't maintain same denominator requirement. To help students: Use visual models (area models, fraction bars) to show different ways to group the same total—3/4 shaded in a rectangle can be grouped as (1 part + 2 parts)/4 or (1 part + 1 part + 1 part)/4, same total but different groupings. Emphasize SAME DENOMINATOR always—when decomposing 3/4, every fraction must have denominator 4 (all fourths). Connect to addition fact families: if you know 1 + 2 = 3 and 1 + 1 + 1 = 3, you can decompose 3/4 those ways.

This question tests 4th grade ability to decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation (CCSS.4.NF.3.b). Decomposing a fraction means breaking it apart into a sum of smaller fractions—like taking 7/12 and writing it as 3/12 + 4/12, or 2/12 + 5/12, or 1/12 + 6/12. The key requirement: all fractions in the decomposition must have the SAME denominator (all twelfths), and the numerators must add up to the original numerator. To decompose 7/12, students must write it as a sum of fractions with denominator 12, where the numerators add to 7. Different decompositions show different ways to partition the 7 parts: 7 twelfths can be grouped as 3+4, or 2+5, or 1+6, etc., all representing the same total. Choice A is correct because both decompositions use denominator 12 throughout, the numerators in first decomposition add to 7: 3+4=7 ✓, the numerators in second decomposition add to 7: 2+5=7 ✓, and the two decompositions are different from each other. Choice D represents order reversal counted as different (3/12+4/12 vs 4/12+3/12), which happens when students think commutative property creates different decomposition. To help students: Use visual models (area models, fraction bars) to show different ways to group the same total—7/12 shaded in a rectangle can be grouped as (3 red + 4 blue)/12 or (2 red + 5 blue)/12, same total but different groupings. Emphasize SAME DENOMINATOR always—when decomposing 7/12, every fraction must have denominator 12 (all twelfths). Show that order doesn't matter (3/12 + 4/12 = 4/12 + 3/12), so don't count these as two different ways.

This question tests 4th grade ability to decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation (CCSS.4.NF.3.b). Decomposing a fraction means breaking it apart into a sum of smaller fractions—like taking 9/8 and writing it as 8/8 + 1/8, or 4/8 + 5/8, or 3/8 + 3/8 + 3/8. The key requirement: all fractions in the decomposition must have the SAME denominator (all eighths, etc.), and the numerators must add up to the original numerator. There are many different ways to decompose the same fraction, as long as these rules are followed. To decompose 9/8, students must write it as a sum of fractions with denominator 8, where the numerators add to 9. Different decompositions show different ways to partition the 9 parts: 9 eighths can be grouped as 8+1, or 4+5, or 3+3+3, etc., all representing the same total. Choice A is correct because both decompositions use denominator 8 throughout, the numerators in first decomposition add to 9 (8 + 1 = 9), the numerators in second decomposition add to 9 (4 + 5 = 9), and the two decompositions are different from each other. This demonstrates understanding that fractions can be decomposed multiple ways while maintaining same denominator and total value. Choice C represents using a whole number instead of a fraction with denominator 8, which happens when students confuse mixed numbers with proper fraction decompositions or don't express wholes as fractions with the same denominator. To help students: Use visual models (area models, fraction bars) to show different ways to group the same total—9/8 (more than one whole) can be grouped as (8 red + 1 blue)/8 or (4 red + 5 blue)/8, same total but different groupings. Emphasize SAME DENOMINATOR always—when decomposing 9/8, every fraction must have denominator 8 (all eighths), including expressing wholes like 1 = 8/8; check that numerators sum correctly: 8 + 1 = 9 ✓, 4 + 5 = 9 ✓, 3 + 3 + 3 = 9 ✓; connect to addition fact families: if you know 8 + 1 = 9 and 4 + 5 = 9, you can decompose 9/8 those ways; show that order doesn't matter (8/8 + 1/8 = 1/8 + 8/8), so don't count these as two different ways; watch for: changing denominators, numerators that don't sum correctly, repeating same decomposition, and counting commutative versions as different.

This question tests 4th grade ability to decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation (CCSS.4.NF.3.b). Decomposing a fraction means breaking it apart into a sum of smaller fractions—like taking 5/12 and writing it as 2/12 + 3/12, or 1/12 + 1/12 + 3/12, or 1/12 + 4/12. The key requirement: all fractions in the decomposition must have the SAME denominator (all twelfths), and the numerators must add up to the original numerator. To decompose 5/12, students must write it as a sum of fractions with denominator 12, where the numerators add to 5. Different decompositions show different ways to partition the 5 parts: 5 twelfths can be grouped as 2+3, or 1+1+3, or 1+4, etc., all representing the same total. Choice A is correct because both decompositions use denominator 12 throughout, the numerators in first decomposition add to 5: 2+3=5 ✓, the numerators in second decomposition add to 5: 1+1+3=5 ✓, and the two decompositions are different from each other (one uses two fractions, the other uses three). Choice B represents numerators that don't sum correctly in the first equation (1+2=3, not 5), which happens when students make arithmetic errors in summing. To help students: Use visual models (area models, fraction bars) to show different ways to group the same total—5/12 shaded in a rectangle can be grouped as (2 parts + 3 parts)/12 or (1 part + 1 part + 3 parts)/12, same total but different groupings. Emphasize SAME DENOMINATOR always—when decomposing 5/12, every fraction must have denominator 12 (all twelfths). Show that using more than two addends is allowed and creates different decompositions.

This question tests 4th grade ability to decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation (CCSS.4.NF.3.b). Decomposing a fraction means breaking it apart into a sum of smaller fractions—like taking 9/8 (an improper fraction) and writing it as 5/8 + 4/8, or 1/8 + 8/8, or 3/8 + 6/8. The key requirement: all fractions in the decomposition must have the SAME denominator (all eighths), and the numerators must add up to the original numerator. To decompose 9/8, students must write it as a sum of fractions with denominator 8, where the numerators add to 9. Different decompositions show different ways to partition the 9 parts: 9 eighths can be grouped as 5+4, or 1+8, or 3+6, etc., all representing the same total. Choice B is correct because both decompositions use denominator 8 throughout, the numerators in first decomposition add to 9: 5+4=9 ✓, the numerators in second decomposition add to 9: 1+8=9 ✓, and the two decompositions are different from each other. Choice A represents numerators that don't sum correctly in the first equation (4+4=8, not 9), which happens when students make arithmetic errors in summing. To help students: Use visual models to show that improper fractions can be decomposed too—9/8 means 9 eighths, which is more than one whole. Emphasize SAME DENOMINATOR always—when decomposing 9/8, every fraction must have denominator 8 (all eighths). Note that 8/8 = 1 whole, so 1/8 + 8/8 = 1/8 + 1 whole = 9/8.

This question tests 4th grade ability to decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation (CCSS.4.NF.3.b). Decomposing a fraction means breaking it apart into a sum of smaller fractions—like taking 5/6 and writing it as 2/6 + 3/6, or 1/6 + 1/6 + 3/6, or 1/6 + 4/6. The key requirement: all fractions in the decomposition must have the SAME denominator (all sixths, etc.), and the numerators must add up to the original numerator. There are many different ways to decompose the same fraction, as long as these rules are followed. To decompose 5/6, students must write it as a sum of fractions with denominator 6, where the numerators add to 5. Different decompositions show different ways to partition the 5 parts: 5 sixths can be grouped as 2+3, or 1+1+3, or 1+4, etc., all representing the same total. Choice A is correct because both decompositions use denominator 6 throughout, the numerators in first decomposition add to 5: 2 + 3 = 5, the numerators in second decomposition add to 5: 1 + 1 + 3 = 5, and the two decompositions are different from each other (different number of addends). This demonstrates understanding that fractions can be decomposed multiple ways while maintaining same denominator and total value. Choice C represents using different denominators, which happens when students don't maintain same denominator requirement or confuse equivalent fractions. To help students: Use visual models (area models, fraction bars) to show different ways to group the same total—5/6 shaded in a rectangle can be grouped as (2 red + 3 blue)/6 or (1 red + 1 green + 3 blue)/6, same total but different groupings. Emphasize SAME DENOMINATOR always—when decomposing 5/6, every fraction must have denominator 6 (all sixths); check that numerators sum correctly: 2 + 3 = 5 ✓, 1 + 1 + 3 = 5 ✓; connect to addition fact families: if you know 2 + 3 = 5 and 1 + 1 + 3 = 5, you can decompose 5/6 those ways; show that order doesn't matter (2/6 + 3/6 = 3/6 + 2/6), so don't count these as two different ways; watch for: changing denominators, numerators that don't sum correctly, repeating same decomposition, and counting commutative versions as different.

This question tests 4th grade ability to decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation (CCSS.4.NF.3.b). Decomposing a fraction means breaking it apart into a sum of smaller fractions—like taking 3/4 and writing it as 1/4 + 2/4, or 1/4 + 1/4 + 1/4. The key requirement: all fractions in the decomposition must have the SAME denominator (all fourths, etc.), and the numerators must add up to the original numerator. There are many different ways to decompose the same fraction, as long as these rules are followed. To decompose 3/4, students must write it as a sum of fractions with denominator 4, where the numerators add to 3. Different decompositions show different ways to partition the 3 parts: 3 fourths can be grouped as 1+2, or 1+1+1, etc., all representing the same total. Choice A is correct because both decompositions use denominator 4 throughout, the numerators in first decomposition add to 3 (1 + 2 = 3), the numerators in second decomposition add to 3 (1 + 1 + 1 = 3), and the two decompositions are different from each other (different number of addends). This demonstrates understanding that fractions can be decomposed multiple ways while maintaining same denominator and total value. Choice B represents using different denominators, which happens when students don't maintain the same denominator requirement or rewrite equivalent fractions without consistent denominators. To help students: Use visual models (area models, fraction bars) to show different ways to group the same total—3/4 shaded in a rectangle can be grouped as (1 red + 2 blue)/4 or three separate 1/4 sections, same total but different groupings. Emphasize SAME DENOMINATOR always—when decomposing 3/4, every fraction must have denominator 4 (all fourths); check that numerators sum correctly: 1 + 2 = 3 ✓, 1 + 1 + 1 = 3 ✓; connect to addition fact families: if you know 1 + 2 = 3 and 1 + 1 + 1 = 3, you can decompose 3/4 those ways; show that order doesn't matter (1/4 + 2/4 = 2/4 + 1/4), so don't count these as two different ways; watch for: changing denominators, numerators that don't sum correctly, repeating same decomposition, and counting commutative versions as different.

This question tests 4th grade ability to decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation (CCSS.4.NF.3.b). Decomposing a fraction means breaking it apart into a sum of smaller fractions—like taking $5/8$ and writing it as $1/8 + 4/8$, or $2/8 + 3/8$, or $1/8 + 1/8 + 3/8$. The key requirement: all fractions in the decomposition must have the SAME denominator (all eighths, all fourths, etc.), and the numerators must add up to the original numerator. To decompose $5/8$, students must write it as a sum of fractions with denominator 8, where the numerators add to 5. Different decompositions show different ways to partition the 5 parts: 5 eighths can be grouped as $1+4$, or $2+3$, or $1+1+3$, etc., all representing the same total. Choice B is correct because both decompositions use denominator 8 throughout, the numerators in first decomposition add to 5: $1+4=5$ ✓, the numerators in second decomposition add to 5: $2+3=5$ ✓, and the two decompositions are different from each other. Choice A represents numerators that don't sum correctly ($1+3=4$, not 5), which happens when students make arithmetic errors in summing. To help students: Use visual models (area models, fraction bars) to show different ways to group the same total—$5/8$ shaded in a rectangle can be grouped as ($1$ red + $4$ blue$)/8$ or ($2$ red + $3$ blue$)/8$, same total but different groupings. Emphasize SAME DENOMINATOR always—when decomposing $5/8$, every fraction must have denominator 8 (all eighths). Check that numerators sum correctly: $1 + 4 = 5$ ✓, $2 + 3 = 5$ ✓, $1 + 1 + 3 = 5$ ✓.

This question tests 4th grade ability to decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation (CCSS.4.NF.3.b). Decomposing a fraction means breaking it apart into a sum of smaller fractions—like taking 6/12 and writing it as 3/12 + 3/12, or 1/12 + 5/12, or 2/12 + 4/12. The key requirement: all fractions in the decomposition must have the SAME denominator (all twelfths), and the numerators must add up to the original numerator. To decompose 6/12, students must write it as a sum of fractions with denominator 12, where the numerators add to 6. Different decompositions show different ways to partition the 6 parts: 6 twelfths can be grouped as 3+3, or 1+5, or 2+4, etc., all representing the same total. Choice A is correct because both decompositions use denominator 12 throughout, the numerators in first decomposition add to 6: 3+3=6 ✓, the numerators in second decomposition add to 6: 1+5=6 ✓, and the two decompositions are different from each other. Choice B represents numerators that don't sum correctly in the first equation (1+4=5, not 6), which happens when students make arithmetic errors in summing. To help students: Use visual models (area models, fraction bars) to show different ways to group the same total—6/12 shaded in a rectangle can be grouped as (3 red + 3 blue)/12 or (1 red + 5 blue)/12, same total but different groupings. Emphasize SAME DENOMINATOR always—when decomposing 6/12, every fraction must have denominator 12 (all twelfths). Check that numerators sum correctly: 3 + 3 = 6 ✓, 1 + 5 = 6 ✓, 2 + 4 = 6 ✓.